21 PRODUCT OF MEASURES 235 system in ^^(l.X) (show that they are all orthogonal to the functions co$2k7rt for k integral and ^>0). Show that /(/) = sgn(sin 2n7rt). (c) For each x =(*)Ł 0 in X, put u(x) = (xn + l)n^0. Show that the measure p, is invariant with respect to u and that u is ergodic with respect to //, (use (a) and Problem 5(c) of Section 13.12). Deduce that lim -(ri(/)+ +rn(/)) = 0 n-voo n almost everywhere with respect to Lebesgue measure ("Borel-Cantelli theorem"). (Observe that rn(\ - f) = -rB(0 for 0 <i t <[ J.) (d) For each finite strictly increasing sequence (nt)1Łt^k of integers ^ 1, show that (e) For each finite sequence (ak)i$k^n of complex numbers, and each real number p > 0, show that r P/2 ("Khintchine's inequality"). (First consider the case where/? = 2/z, with h an integer 2:1, by using (d). For 2h 2<p<:2h, use problem 12(e) of Section 13.11.) (f) With the notation of (e), show that (n \l/2 /! n fc.i / ^ Jo *=1 (use (e) for p = 4, and Holder's inequality for p = | and q 3). 11. (a) The hypotheses are those of Section 13.17, Problem 2, and in addition the functions / are assumed to be real. Let xh-+j(x) be a ju-measurable mapping of X into the set I = {1, 2,..., /?}, and let k\*w(k) be an increasing function on I with values >0. Then the mapping (s9 w)h~>Ky(S)(5<, u)lw(j(s)) of X x X into R is p, Ž /it- measurable. For each /^-measurable subset A of X, show that where h(s, t) = inf(j(s)J(t)). (Express the square of the integral on the left as a double integral, so as to reduce the problem to the evaluation of a triple integral over A x A x X, and use the fact that the / are orthogonal.) Show next that r |K*(M,(j, 01 J (split up A x A into two measurable sets, defined respectively by j(s) <j(t) and J($) Sy'(') for (s, 0 e A x A and use the fact that Kn(s, r) = Kn(f, s)). (b) Suppose that there exists an increasing sequence n \-*w(ri) of numbers >0 and a /x-measurable subset A of X such that |HnC?)| ^ c yv(n) for all ^ e A and all n ^ 1, where c> 0 is a constant. Show that, for each function ge JS?Ł(X, /A), the sequence (sn(g)(t))lw(ri) is bounded for almost all t e A, (Consider the increasing sequence of u-integrable functions vn(t) sup (j*(^)(/))/w(A:), and show that the sequence off X let un be the measure for which ju,n(0) = /^n(l) = i, and let a = Ž un be the